Equivalent sets. Are they interchangeable?

Question

Freshman question, really, but the more I think about it, the more I doubt.

Suppose that two sets belong to the same equivalence class. Are they in effect interchangeable? (I understand that there is no axiom of `interchangeability' in the definition of an equivalence relation.)

For example, consider the equivalence class of all people who are 30 years old. This equivalence class contains both men and women who are 30; and men and women are different 'objects' if I may say. Yet if I consider the class of people who are 30 for some analysis, it does not matter if I pick a man or a woman. They are interchangeable as long as what matter is their age.

I just wonder if this is characteristic of all equivalence classes one can encounter in mathematics.

Answer 1

The answer to your question is in the question, here:

They are interchangeable as long as what matter is their age.

If all that matters in any particular context is the condition that specifies the equivalence relation then any representative will do.

Answer 2

I may be just rephrasing some of the answers, but I like to think that an equivalence relation is simply a problem specific notion of equality; the relation tries to summarize all the important properties that are relevant to the problem in question. In this sense, if any two objects are equivalent they can be used interchangeably without consequence.

To exemplify, you can consider that a sphere is equivalent to a cylinder if all that matters to your problem is purely topological. As long as distances start to matter, this topological notion of 'equality' is not suitable anymore you should look for a more stringent notion of 'equality'.